Chalmers Conferences, 9th European Conference on Mathematical and Theoretical Biology

A simplified, geometrical approach to the PKPD of inhaled bronchodilators
Claudio Roberto Gaz, Andrea De Gaetano, Simona Panunzi, George Cremona, Beverley Patterson

Last modified: 2014-03-31


Administration of drugs by inhalation is a common therapeutic approach for a number of medical conditions, particularly for asthma. Assessing in vivo aerosol lung deposition, together with the ensuing pharmacokinetics (PK)  and pharmacodynamics (PD), is still a challenging task: the lung is anatomically complex, with a high degree of inter-individual variability, as well as intra-individual variability over time. Furthermore, the deposition of an inhaled dose of compound is affected by many factors, such as device construction, formulation composition, particle aerodynamic diameter and the subject's ability to generate inhaled airflow.
In order to describe the fate of compounds administered by inhalation of an aerosolized cloud of droplets of a liquid suspension, or of a dry powder formulation, two diametrically opposite approaches can be envisioned.

A first approach consists in modelling the breathing system, including the blood to tissue distribution of bronchoactive substances, as a collection of functional blocks (compartments), where each block approximately represents anatomical structures of homogeneous characteristics, and where the drug is assumed to be instantaneously well-stirred: this gives rise to a system of (typically) nonlinear ordinary differential equation, with the caveat that not all compartments have, with good approximation, only a time-dependent behaviour. In fact, the repeatedly branching bronchial tree determines a kinetic disposition that is not only a function
of time, but is also a function of the hierarchical arrangement of the bronchi themselves.
A second approach is to conduct a computational fluid dynamics (CFD) study, in which the breathing system of a given subject, from larynx to lower bronchioles, is geometrically modelled in three-dimensional space, typically on the basis of CT scans or other medical images. By means of a numerical solution of the Navier-Stokes equations, the motion through the airways of the drug-suspension droplets or particles is closely approximated and the regions where the droplets impact the internal surface of the bronchial tree may be directly computed. This approach is however very computationally intensive, and each three-dimensional model used to represent the bronchial tree is strictly related to a given individual at a given moment, so that very limited generalization is possible, few subjects can be analyzed, and a population approach is not feasible.
The approach explored in the present work walks a middle way between the above described methods. Instead of modelling the bronchial region as a standardized, homogeneous compartment (with one or few ordinary differential equations), a more mechanistic view is obtained by modelling it as a partial differential process in one spatial variable, where many crucial PK and PD features depend on the spatial dimension. This representation is much simpler and faster than CFD and may be employed in clinical trials.
The present model couples three ODEs  (for Gut, Plasma and Urine) and 2 PDEs (for Bronchial Mucosa and Bronchial Muscle layers) in order to describe the PK. The PD is obtained by modelling features depending on the spatial dimension, the depth along the bronchial tree, such as the number of branches (and thus the total bronchial surface at a given depth), the number of beta-2 adrenergic receptors and the offered airways resistance to the flow of air. In particular, bronchial resistance is modeled in accord with empirical observations from the literature, and expiratory flow is then obtained from Ohm's law. The local effect of the bronchodilator at a given bronchial depth is then simulated by increasing the the bronchial diameter depending on the number of receptors and the quantity of drug active at the given bronchial depth.
The model is validated by reproducing the time trend of the Forced Expiratory maneuvers in broncho-constricted subjects.


PDEs; mathematical modelling; respiratory system; bronchodilators; COPD; asthma; FEV1